Rhombidodecadodecahedron

In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices.[1] It is given a Schläfli symbol t0,2{​5⁄2,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.

Rhombidodecadodecahedron
TypeUniform star polyhedron
ElementsF = 54, E = 120
V = 60 (χ = −6)
Faces by sides30{4}+12{5}+12{5/2}
Wythoff symbol5/2 5 | 2
Symmetry groupIh, [5,3], *532
Index referencesU38, C48, W76
Dual polyhedronMedial deltoidal hexecontahedron
Vertex figure
4.5/2.4.5
Bowers acronymRaded
3D model of a rhombidodecadodecahedron

Cartesian coordinates

Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of

(±1/τ2, 0, ±τ2)
(±1, ±1, ±√5)
(±2, ±1/τ, ±τ)

where τ = (1+√5)/2 is the golden ratio (sometimes written φ).

It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the icosidodecadodecahedron (having the pentagonal and pentagrammic faces in common) and the rhombicosahedron (having the square faces in common).


convex hull
Rhombidodecadodecahedron
Icosidodecadodecahedron

Rhombicosahedron
Compound of ten triangular prisms
Compound of twenty triangular prisms

Medial deltoidal hexecontahedron
Medial deltoidal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 54 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU38
dual polyhedronRhombidodecadodecahedron
3D model of a medial deltoidal hexecontahedron

The medial deltoidal hexecontahedron (or midly lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.

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See also

References
  1. Maeder, Roman. "38: rhombidodecadodecahedron". MathConsult.


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